Periodic Solutions - Rayleigh’s Equation
The British mathematical physicist Lord Rayleigh (John William Strutt, 1842–1919) introduced an equation of the form
mx''+kx=ax'-b(x')3
to model the oscillations of a clarinet reed.
The outward and inward spiral trajectories converge to a “limit cycle” solution that corresponds to periodic oscillations of the reed. The period (and hence the frequency) of these oscillations can be measured on a solution curve plotted. This period of oscillation depends only on the parameters m, k, a, and b in and is independent of the initial conditions.
Phase portrait for m=a=b=k=1
A typical tx-solution curve (here with initial conditions x(0)=0.01, x′(0)=0):
Your Turn
Choose your own parameters (perhaps the least four nonzero digits in your student ID number), and plot trajectories and solution curves as in the figures above. Change one of your parameters to see how the amplitude and frequency of the resulting periodic oscillations are altered.
Rayleigh’s Equation